1. **State the problem:** We have a right triangle ABC with a right angle at C, angle A = 65°, and hypotenuse AB = 21 units. We need to find the lengths of sides a (BC) and b (AC). 2. **Recall the trigonometric relationships:** In a right triangle, - $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ - $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$ 3. **Identify sides relative to angle A:** - Opposite side to angle A is $a = BC$ - Adjacent side to angle A is $b = AC$ - Hypotenuse is $c = AB = 21$ 4. **Calculate side a using sine:** $$a = c \times \sin(65^\circ) = 21 \times \sin(65^\circ)$$ Using $\sin(65^\circ) \approx 0.9063$, $$a = 21 \times 0.9063 = 19.0323$$ 5. **Calculate side b using cosine:** $$b = c \times \cos(65^\circ) = 21 \times \cos(65^\circ)$$ Using $\cos(65^\circ) \approx 0.4226$, $$b = 21 \times 0.4226 = 8.8746$$ 6. **Final answers:** - $a \approx 19.03$ - $b \approx 8.87$ These are the lengths of the legs of the right triangle.